Abstracts
Yiping Ma (Northumbria University Newcastle)
Traveling edge waves in photonic graphene
Photonic graphene, namely an honeycomb array of optical waveguide, has attracted much interest in the optics community. In recent experiments it was shown that introducing edges and suitable waveguides in the direction of propagation, unidirectional edge wave propagation at optical frequencies occurs in photonic graphene. The system is described analytically by the lattice nonlinear Schrodinger (NLS) equation with a honeycomb potential and a pseudo-magnetic field. In certain parameter regimes, these edge waves were found to be nearly immune to backscattering, and thus exhibit the hallmarks of (Floquet) topological insulators.
This talk addresses the linear and nonlinear dynamics of traveling edge waves in photonic graphene, using a tight-binding model derived from the lattice NLS equation. Two different asymptotic regimes are discussed, in which the pseudo-magnetic field is respectively assumed to vary rapidly and slowly. In the presence of nonlinearity, nonlinear edge solitons can exist due to the balance between dispersion and nonlinearity; these edge solitons appear to be immune to backscattering when the dispersion relation is topologically nontrivial. A remarkable effect of topology in bounded photonic graphene will also be demonstrated: the edge modes are found to exhibit strong transmission (reflection) around sharp corners when the dispersion relation is topologically nontrivial (trivial).
Fabio Briscese (Northumria University Newcastle)
Theoretical foundations of the Schroedinger method for the formation of large scale structures in the Universe
It has been shown that the formation of large scale structures (LSS) in the universe can be described in terms of a Schroedinger-Poisson system. This procedure, known as Schroedinger method, has no theoretical basis, but it is intended as a mere tool to model the N-body dynamics of dark matter halos which form LSS. Furthermore, in this approach the ``Planck constant” in the Schroedinger equation is just a free parameter. In this seminar I will discuss a possible derivation of the Schroedinger method which is based on the stochastic quantization introduced by Nelson, and on the Calogero conjecture. Moreover, I will discuss how the Calogero conjecture allows to estimate the value of the effective Planck constant.
Oleg Kirillov (Northumbria University Newcastle)
Diabolical and exceptional points in the families of non-Hermitian matrices
We consider a complex perturbation of multiparameter families of real symmetric and Hermitian matrices and study unfolding of conical singularities of eigensurfaces at diabolical points into new singular surfaces such as the conical wedge of Wallis. As a physical application, singularities of dispersion surfaces in crystal optics are studied. Asymptotic formulas for the metamorphoses of these surfaces depending
on properties of a crystal are established and discussed. Singular axes for general crystals with weak absorption and chirality are found. An explicit condition distinguishing the absorbtion-dominated and chirality-dominated crystals is established in terms of components of the inverse dielectric tensor. Finally, we turn to the question of approximate computation of the geometric phase along a path surrounding either DP or EP-set in the parameter space by means of the perturbation of eigenvectors at the degeneracies
Andrea Pizzoferrato (University of Warwick)
Lower current large deviations for zero-range processes on a ring
Non-equilibrium statistical mechanics is a broad field of research covering many different phenomena. In this context Interacting Particle Systems (IPS) are a well established class of minimal models, where particles jump on a lattice according to local jump rates. This talk considers one particular IPS called the Zero Range Process (ZRP), which is characterized by jump rates depending on the number of particles on the departure site only. Remarkably, for certain choices of the transition rate function, the model exhibits “condensation”, that is a finite fraction of the particles of the system concentrates on the same site. We focus on totally asymmetric dynamics with periodic boundary conditions in one dimension, and study the large deviations of the particle current. Lower large deviations can be realized by phase separated states of high and low density regions, which may degenerate into condensed profiles for condensing ZRPs. We establish a dynamic phase transition related to this crossover and derive the rate function for the current for a large class of ZRPs. The results presented can be found in more detail here: https://arxiv.org/abs/1611.03729. This is a joint work with Paul Chleboun and Stefan Grosskinsky.
Francesco Demontis (University of Cagliari)
Soliton solutions for the classical continuous Heisenberg ferromagnet equation
Many nonlinear differential equations can be solved via the Inverse Scattering Transform (IST). In this talk, after a brief introduction of the IST, we present a rigorous formulation of the IST for the classical continuous Heisenberg ferromagnet (HF) equation. This formulation is based on a new triangular representation for the Jost solutions, which in turn allows to establish the asymptotic behaviour of the scattering data for large values of the spectral parameter. A new, general, explicit multi-soliton solution formula, amenable to computer algebra, is obtained by means of the matrix triplet method, producing all the soliton solutions for the HF equation and allowing their classification and description.
This talk is based on a joint work with M. Sommacal and S. Lombardo (Northumbria University, Newcastle) and C. van der Mee and F. Vargiu (University of Cagliari).
Mario Angelelli (University of Salento)
If it is complex, make it simplex: A tropical approach to statistical physics
The concepts of partition function and free energy lie at the root of statistical physics and are now pervasive in many branches of sciences. This talk is meant to explore some algebraic and geometric aspects of these fundamental objects, with emphasis on their role in the connection of micro- and macro-physics. Firstly, some geometric realizations of these connections will be briefly presented. Then, main attention will be paid to the issues of composition and dominance, which are pivotal in statistical physics and will be expressed through tropical algebras. A basic introduction on these structures will be given in order to discuss the tropical limit of statistical models. Concrete examples will be provided, with particular focus on highly (exponentially) degenerate models and their physical manifestations (residual entropy, limiting temperatures). This scheme is formalized in a micro-macro correspondence, where microphysics is related to a macroscopic description within the composition rules given by tropical structures. Some physical consequences of this correspondence are discussed, i.e. relations with ultrametrics, non-equilibrium and tropical equilibrium, dependence on the choice of ground energy.
Antonio Moro (Northumbria University Newcastle)
Dressing networks: towards an integrability approach to collective and complex phenomena
A large variety of real world systems can be naturally modelled by networks, i.e. graphs whose nodes represent the components of a system linked (interacting) according to specific statistical rules. A network is realised by a graph typically constituted of a large number of nodes/links. Fluid and magnetic models in Physics are just two among the many classical examples of systems which can be modelled by using simple or complex networks. In particular "extreme" conditions (thermodynamic regime), networks, just like fluids and magnets, exhibit a critical collective behaviour intended as a drastic change of state due to a continuous change of thermodynamic parameters.
Using an approach to thermodynamics, recently introduced to describe a general class of van der Waals type models and magnetic systems in mean field approximation, we analyse the integrable structure of corresponding networks and use the theory of nonlinear conservation laws to provide an analytical description of the system outside and inside the critical region.
Vladimir Novikov (Loughborough University)
Multi-dimensional integrable systems: from dispersionless to soliton equations
In this talk we will consider the problem of studying, detecting and classifying integrable 2+1-dimensional equations. Our approach is based on the observation that dispersionless limits of integrable systems in 2 + 1 dimensions possess infinitely many multi-phase solutions coming from the so-called hydrodynamic reductions. We develop a novel perturbative approach to the classification problem of dispersive equations. Based on the method of hydrodynamic reductions, we first classify integrable quasilinear systems which may (potentially) occur as dispersionless limits of soliton equations in 2 + 1 dimensions. To reconstruct dispersive deformations, we require that all hydrodynamic reductions of the dispersionless limit are inherited by the corresponding dispersive counterpart. This procedure leads to a complete list of integrable third and fifth order equations, which generalize the examples of Kadomtsev-Petviashvili, Veselov-Novikov and Harry Dym equations as well as integrable Davey-Stewartson type equations, some of which are apparently new. We also extend the technique to differential-difference and fully discrete integrable systems in 3D.
Arseni Goussev (Northumbria University Newcastle)
Rotating Gaussian wave packets in weak external potentials
Among many motivations to study the time evolution of quantum matter-wave packets two are particularly noteworthy. First, localized wave packets provide the most natural tool for investigating the correspondence between quantum and classical motion. Indeed, while the center of a propagating wave packet traces a trajectory, a concept essential in classical mechanics, its finite spatial extent makes quantum interference effects possible. Second, wave packets may be used as basis functions. That is, any initial state of a quantum system can be represented as a superposition of a number, finite or infinite, of localized wave packets. Thus, an analytical understanding of how each individual wave packet moves through space offers a way to quantitatively describe the time-evolution of an arbitrary, often complex, initial state. Despite a large body of literature on quantum wave packet dynamics, the subject is by no means exhausted.
In a recent paper [1], Dodonov has addressed the time evolution of nonrelativistic two-dimensional Gaussian wave packets with a finite value of mean angular momentum (MAM). The value is the sum of the "external" MAM, related to the center of mass motion, and the "internal" MAM, resulting from the rotation of the wave packet around its center of mass. Among many interesting features of such wave packets is the effect of initial shrinking of packets with strong enough coordinate-momentum correlation.
In my talk, motivated by Ref. [1], I will consider the propagation of a localized two- or three-dimensional rotating Gaussian wave packet in the presence of a weak external potential. The particular focus will be on the time evolution of the internal MAM of the moving wave packet. I will derive, using a semiclassical approximation of the eikonal type, an explicit formula that gives the value of the internal MAM as a function of the propagation time, parameters of the initial wave packet and the external potential. An example physical scenario, in which a two-dimensional particle traverses a tilted ridge barrier, will be considered in full detail. In particular, it will be shown how an initially uncorrelated, rotation-free wave packet may, upon a collision with the potential barrier, acquire a finite internal MAM, and how the magnitude and direction of the MAM are determined by the aspect ratio and orientation of the incident wave packet.
[1] V. V. Dodonov. "Rotating quantum Gaussian packets," J. Phys. A: Math. Theor. 48, 435303 (2015).
Gregory Morozov (University of the West of Scotland)
Exactly solvable Hill's equations
Remy Dubertrand (University of Liege)
Scattering theory for walking droplets in the presence of obstacles
Walking droplets that are sustained on the surface of a vibrating liquid, have attracted considerable attention during the past decade due to their remarkable analogy with quantum wave-particle duality. This was initiated by the pioneering experiment by Y. Couder and E. Fort in 2006, which reported the observation of a diffraction pattern in the angular resolved profile of droplets that propagated across a single slit obstacle geometry. While the occurrence of this wave-like phenomenon can be qualitatively traced back to the coupling of the droplet with its associated surface wave, a quantitative framework for the description of the surface-wave-propelled motion of the droplet in the presence of confining boundaries and obstacles still represents a major challenge. This problem is all the more stimulating as several experiments have already reported clear effects of the geometry on the dynamics of walking droplets.
Here we present a simple model inspired from quantum mechanics for the dynamics of a walking droplet in an arbitrary geometry. We propose to describe its trajectory using a Green function approach. The Green function is related to the Helmholtz equation with Neumann boundary conditions on the obstacle(s) and outgoing conditions at infinity.
For a single slit geometry our model is exactly solvable and reproduces some of the features observed experimentally. It stands for a promising candidate to account for the presence of boundaries in the walker’s dynamics.
Martin Brinkmann (Universitat de Saarlandes)
Wettability controls fluid transport in particulate and permeable media
While the impact of pore shape and pore size distributions on immiscible displacement has been widely studied, only a few previous experiments and simulations have addressed the effect of pore-scale wettability. In this talk, I will present examples of two and three dimensional experimental model system where both geometry and wettability are controllable factors. Optical microscopy and x-ray tomography are employed to monitor and track the distribution of fluids down to the
level of single pores. To gain further insight into the underlying pore-scale processes, we have developed a stochastic rotation dynamic model for multi-phase flows that account for wall wettability. This allows us to assess and validate experiments of fluid transport in Hele-Shaw cells with cylindrical obstacles or random packings of spherical beads.
Benoit Huard (Northumbria University Newcastle)
Title: Periodic solutions in delay equations with application to glucose regulation rhythms
Delay-differential equations represent a class of dynamical systems which offer an enhanced modelling of physical and biological processes through the incorporation of time delays. These typically represent mechanisms, physical or physiological, which incur a delayed response within the system. The resulting dynamical system is infinite-dimensional and one can very rarely obtain closed-form exact solutions. Nonetheless, characterising bifurcations in these models leads to precise conditions ensuring that the system enters an oscillatory regime.
In this talk, I will be focusing on the description of the so-called ultradian rhythms which are crucial for the appropriate regulation of blood glucose. Two models, with one and two delays respectively, are studied to characterise the periodic solutions and the effect of deficiencies (Type 1 and Type 2 diabetes) on the production of an oscillatory regime. Pathways for restoring these oscillations are discussed. Approximate periodic solutions are obtained using a perturbative analysis. Finally, a new model making use of a state-dependent delay is introduced to assess the efficacy of emergency mechanisms under high glucose levels.
Alexander Veselov (University of Loughborough)
Markov triples: arithmetic, geometry and dynamics
Markov triples are the integer solutions of the Markov equation x^2+y^2+z^2=3xyz, which surprisingly appeared in many areas of mathematics: initially in classical number theory, but more recently in hyperbolic and algebraic geometry, the theory of Teichmueller spaces, Frobenius manifolds and Painleve equations.
I am planning to explain some of these magical relations and properties of Markov triples, including recent results about their growth found jointly with K. Spalding.
Jonathan Halliwell (Imperial College London)
How the Quantum Universe became Classical
Quantum theory has been spectacularly successful in its description of the microscopic realm and there is not one shred of experimental evidence that suggests that its basic structure is incorrect in any way. Yet despite being the fundamental theory of matter, it is not the theory of the ordinary everyday world of our experience and indeed some of its features such as the existence of superposition states and entanglement defy intuition in a truly profound way. This is because the macroscopic world and our everyday experience is best described by the classical mechanics of Newton and the laws of thermodynamics, theories utterly different to quantum theory. This then leads to a very interesting question. If macroscopic objects are made of atoms, and atoms are described by quantum theory, how do large collections of atoms come to be described by classical physics? In brief, how does classical physics emerge from quantum theory at sufficiently large scales? In this talk I outline in simple terms how this transition from quantum to classical physics takes place, with an emphasis on simple physical ideas and pictures, and not elaborate mathematics. It includes a simple account of the decoherent histories approach to quantum theory which is particularly suited to this task.
Sven Gnutzmann (University of Nottingham)
Title: Quantum signatures of chirality in chaotic quantum systems
We revisit the model of two coupled spins by Peres and Feingold. Thirty years ago this was a paradigm system of quantum chaos. We show that this model can be used as a paradigm to understand universal features in the density of states for the novel symmetry classes in The Altland-Zirnbauer's ten-fold symmetry classification. Reducing the system to invariant subspaces reveals that the same classical system may belong to different symmetry classes depending on the spin quantum numbers. Moreover a nonzero topological quantum numbers can be found in various cases. By varying coupling constants the system makes a transition from integrable to chaotic and we show numerically that the repulsion of the quantum energy eigenvalues close to E=0 is consistent with Gaussian random matrix predictions for the corresponding combination of symmetry class and topological quantum number. No proper disorder average is necessary in the sense that we can keep the classical dynamics fixed and average over different quantum representations (spin quantum numbers).
Mon 12th - 3pm - Andrew Hone (University of Kent)
Title: Peakonomics
The Camassa-Holm equation was originally proposed as a model for shallow water waves. It is also an integrable partial differential equation which appeared in earlier work of Fokas and Fuchssteiner on hereditary symmetries, and as well as having a (bi-)Hamiltonian formulation, it can also be interpreted as a geodesic flow on the group of diffeomorphisms with respect to a suitable Sobolev metric. However, perhaps its most interesting feature is that, in the dispersionless case, its solitons are weak solutions - "peakons" - with a discontinuous derivative at isolated points.
After a brief review of peakons in integrable systems, a model for urban growth due to Krugman is presented, which was proposed as a mechanism for the formation of edge cities (that is to say, regions of concentrated economic activity). We show that Krugman's model admits exact measure-valued solutions, with an associated quantity, called the market potential, being given by a superposition of peakons with two different widths. Consideration of the dynamical system for this peakon interaction leads to a conjecture on the form of a global attractor in Krugman's model.
Richard Bertram (Florida State University)
Title: How Simple Concepts from Dynamics Can Drive Biological Experiments
Sometimes knowing a few concepts from dynamics can take you a long way. In this presentation I will demonstrate this claim by describing some research on the origin of pulsatile insulin secretion from pancreatic islets of Langerhans. This research has largely been driven by mathematical insights, which include some well known and very useful concepts about dynamical systems. I hope to relay the message that mathematical theory, computer simulations, and experimental studies can be combined very effectively to increase our understanding of complex biological phenomena.
Matteo Sommacal (Northumbria University)
Title: Linear stability analysis of integrable partial differential equations
Analytical methods of the theory of integrable partial differential equations (PDEs) in 1+1 dimensions have been successfully applied to investigate a number of wave propagation models of physical interest. This talk shows how to address the issue of linear stability of wave solutions by means of these methods. By imitating the standard steps followed when dealing with non-integrable equations, we show how the linear stability of solutions of integrable PDEs can be effectively analysed by using their Lax representation. The most relevant application of this scheme is the analysis of the background continuous wave solution and of the conditions for the existence of rogue waves. The talk is based on work done in collaboration with Antonio Degasperis, University of Rome La Sapienza, Rome, Italy and Sara Lombardo, Northumbria University, Newcastle upon Tyne, UK.
Francesco Giglio (Northumbria University)
Title: Integrable Nematics
Liquid crystals are considered to be, after gases, liquids and solids, the forth natural state of matter on Earth, as terrestrial free standing plasmas have not been yet observed.
Nematic liquid crystals (nematics in short) are characterised by the fact that they occur in two macroscopic phases: isotropic and anisotropic phases. While in the former each molecule is oriented randomly in space, in the latter molecules tend to align along special directions giving rise to a macroscopic anisotropy of the material.
Inspired by a recent formulation of Thermodynamics based on the theory of nonlinear conservation laws, we propose a novel approach to the Statistical Mechanics of nematics that relies on the study of differential identities for the partition function.
We show how exact equations of state of a discrete version of the Maier Saupe model can obtained from the solution of a suitable initial value problem for a partial differential equation.
Aim of the talk is to illustrate the approach and discuss the rich Thermodynamics features encoded in equations of state so obtained.
Rodrigo Ledesma-Aguilar (Northumbria University)
Title: Snap evaporation on smooth topographies
The evaporation of droplets on solid surfaces is important for a broad range of applications that include ink-jet printing, surface cooling, and nano- and micro-structure assembly. Despite its apparent simplicity, the precise configuration of an evaporating droplet on a solid surface has proven notoriously difficult to predict and control. This is because droplet evaporation typically proceeds as a ‘stick-slip’ sequence, which is a combination of pinning and de-pinning events of the droplet edge dominated by the static friction, or ‘pinning’, caused by microscopic structure of the solid surface. Here we show how smooth, pinning-free, solid surfaces of non-planar topography give rise to a different process which we dub snap evaporation. During snap evaporation the morphology of an evaporating droplet follows a reproducible sequence of steps, where the liquid-gas interface is quasi-statically reduced by mass diffusion until it undergoes an out-of-equilibrium snap. Such events are triggered by bifurcations of the equilibrium droplet configuration promoted by the underlying surface topography. Because the evolution of droplets during snap evaporation is controlled by the geometry of the solid, and not by microscopic surface roughness, our ideas are amenable to programmable surfaces which manage evaporation in heat and mass transfer applications.
Adam Bridgewater (Northumbria University Newcastle)
Title: Mathematical investigation of diabetically impaired ultradian oscillations
in the glucose-insulin regulatory system
The accuracy of the oscillatory nature of the glucose and insulin blood levels constitutes an important indicator of healthy regulation [4]. In this contribution, we study a mathematical model which incorporates two physiological delays, as well as parameters representing diabetic deficiencies, in order to investigate the impact of a fault in the glucose utilisation on the production of ultradian oscillations in the glucose-insulin system. The delays represent the hepatic glucose production and the time necessary for the release of insulin, and have been shown to be at least partly responsible for the oscillatory nature of the regime [2,3]. A numerical study of the non-linear mathematical model is performed to characterise the onset of the oscillations, and perturbations of the periodic solutions are used to investigate the amplitude and frequency of the oscillations. Through use of linear stability analysis and bifurcation theory, pathways to restoring appropriate cyclic regulation are mathematically described [1]. Our goal is to provide measurable indicators of deficiency in the system and a more thorough description of the contribution of insulin treatments to the reintroduction of an oscillatory behaviour which is crucial for the design of a control algorithm which could then be implemented into an insulin control system.
[1] Huard, B., Bridgewater, A., Angelova, M. (2017), J Theor Biol, 418, pp.66-76.
[2] Li, J., Kuang, Y., Mason, C. (2006), J Theor Biol 242(3), 722-735.
[3] Sturis, J. et al (1991), AM J Physiol-Endoc M 260(5), E801-E809.
[4] Tolic, I. et al (2000), J Theor Biol 207(3), 361-375.
Cosimo Gorini (University of Regensburg)
Title: Magnetoconductance in (curved) topological insulator nanowires
We investigate quantum transport in 3D topological insulator nanowires in external electric and magnetic fields. The wires host topologically non-trivial surface states wrapped around an insulating bulk, and a magnetic field along the wire axis leads to Aharonov-Bohm oscillations of the conductance. Such oscillations have been observed in numerous systems and signal surface transport, though alone cannot prove its topological nature. Furthermore, it is not known how they are affected by the wire specific geometry - never perfectly cylindrical as in standard theoretical models.
We thus focus on two issues: (i) An accurate modelling of surface transport in gated, strained HgTe nanowires, accompanying experimental measurements performed by our collaborators; (ii) A theoretical study of magneto-conductance through shaped (tapered, curved) nanowires. The nanowire non-constant radius leads to novel quantum transport phenomena. Notably, it implies a competition between effects due to quantum confinement
and to a spatially varying enclosed magnetic flux, as well as offering the possibility of studying quantum Hall physics in curved space.
Sergiy Shelyag (Northumbria University)
Title: Numerical Models for solar plasmas
Due to their gravitational stratification and rotation, solar and stellar plasmas represent a complicated and interesting object for numerical modelling. Orders-of-magnitude variations in thermodynamic and magnetic parameters over short spatial scales impose additional stiffness into the system of partial differential equations describing the plasma behaviour. A variety of physical effects, which affect plasma parameters and act on a large range of spatial and temporal scales, require robustness and computational effectiveness of the numerical methods employed in solar plasma modelling. In my presentation I will cover my own experience in development of numerical models for simulations of solar plasmas. I will describe numerical methods for simulations of waves, flows and radiation in solar plasmas and present an overview of the results, applicable in solar physics.
Ciro Semprebon (Northumbria University)
Title: Numerical simulations of ternary multiphase-multicomponent systems
Systems containing two or more fluid phases and one gas phase are of special practical interest. Oil lubricated surfaces offer a variety of unique properties such as superior non-wetting, ice-repellent and robust drag-reduction. In engines, collision merged droplets of oil and water droplets can increase the effective burning rate. Despite the growing number of applications and the advancing experimental techniques, accurate and flexible models that can predict complex interface dynamics of ternary systems with significant density ratio are lacking. In this talk I will introduce two novel free energy Lattice Boltzmann models and show applications in Liquid Infused Substrates and collisions of immiscible drops.
Kuo-Long Pan (University of Leeds)
Title: On the impact of binary droplets with identical and distinct liquids
Collision between droplets plays a critical role in various subjects of science and technology. For instance, the collisions of oil droplets are the essential elements in liquid-fueled burning devices such as aircraft/rocket combustors and automobile engines. They are the key processes underlying spray combustion, concerning the interactions between different groups of droplets or between droplets and the confinement, which dominate the subsequent distributions of mass, momentum, and energy that consequently determine the behaviors and performance of the burners. The typical outcomes of droplet collisions can be categorized into coalescence, bouncing, temporary coalescence and separation followed by satellite droplets, and splattering, with increase of impact inertia relative to surface tension. In addition to the significance in aerospace applications, the impact dynamics of droplets are important in many other fields as well, e.g. raindrop formation and aerosol phenomena, ink-jet printing technology, firefighting strategy, nuclear power generation, spray painting and coating techniques, etc. To understand the mechanisms and further control the collision outcomes, we have studied the fundamental structures in terms of experimental, analytical, and computational approaches, associated with various physical schemes and length scales. Comprehension of the principles can be extended to other intriguing areas specifically micro/nano multiphase fluidics that has been fervidly investigated recently due to its immediate relevance to biological/medical techniques and industry. In addition to droplets made of identical liquid, we have also studied the collisions between two droplets made of different liquids. In this talk I will introduce some of these studies and discuss the elementary mechanisms.
Biography
Prof. Kuo-Long Pan graduated from National Tsing Hua University in 1994 (B.S.) and 1996 (MS) in Power Mechanical Engineering, and received PhD from Princeton University in 2004 in Mechanical and Aerospace Engineering. Since 2013 he has been a full professor in Department of Mechanical Engineering, National Taiwan University. His research interests lie in fluid physics, combustion and energy, computational fluid dynamics and propulsion. He is an Associate Fellow of American Institute of Aeronautics and Astronautics (AIAA) and a member of American Physical Society (APS). He has received several awards including a Best Paper Award of AIAA (2005)
Fabian Maucher (University of Durham)
Title: Two Examples of Self-organization
Self-organization represents one of the most striking phenomena of complex systems. In this talk I present two
examples of self-organization. The rst example presents dynamics of spiral waves governed by reaction-diusion
equations. There are a range of chemical, physical and biological excitable media that support spiral wave vortices.
Examples include the Belousov-Zhabotinsky redox reaction, the chemotaxis of slime mould and action potentials
in cardiac tissue. There are types of reaction-diusion equations that give rise to strong short-ranged repulsive
interactions between vortex strings [1]. A Biot-Savart construction to initialize a given knot as a vortex string in
the FitzHugh-Nagumo equations is presented [see Fig. 1(a)]. The subsequent evolution gives rise to self-organized
untangling of vortex strings [2]. Light-propagation in media with competing nonlocal nonlinearities represents the
second example of self-organization [3]. Such system can be realized in a gas of thermal alkali atoms. Apart from
spatial soliton formation, the dierent length scales of the nonlocality can give rise to lamentation and subsequent
self-organised hexagonal lattice formation in the beam prole [see Fig. 1(b)], akin to the superuid-supersolid phase
transition in Bose-Einstein condensates.
Oleg Chalykh (University of Leeds)
Title: Dunkl-Cherednik operators and quantum Lax pairs
I will explain a direct link between Dunkl operators and quantum Lax pairs for the elliptic Calogero--Moser systems. This produces Lax pairs with a spectral parameter, which in the classical limit reduce to the well-known Krichever's Lax pair and its analogues for other root systems. Time permitting, I will outline a similar construction for the elliptic Ruijsenaars-Schneider system and its variants. The Dunkl operators in that case are replaced by their q-analogues, also known as Cherednik operators.
Alessia Annibale (King's College London)
Title: Cell reprogramming modelled as transitions in a hierarchy of cell cycles
We construct a model of cell reprogramming (the conversion of fully differentiated cells to a state of pluripotency, known as induced pluripotent stem cells, or iPSCs) which builds on key elements of cell biology i.e. cell cycles and cell lineages. Although reprogramming has been demonstrated experimentally, much of the underlying processes governing cell fate decisions remain unknown. This work aims to bridge this gap by modelling cell types as a set of hierarchically related dynamical attractors representing cell cycles. Stages of the cell cycle are characterised by the configuration of gene expression levels, and reprogramming corresponds to triggering transitions between such configurations. Two mechanisms were found for reprogramming in a two level hierarchy: cycle specific perturbations and a noise-induced switching. The former corresponds to a directed perturbation that induces a transition into a cycle state of a different cell type in the potency hierarchy (mainly a stem cell) whilst the latter is a priori undirected and could be induced, e.g., by a (stochastic) change in the cellular environment. These reprogramming protocols were found to be effective in large regimes of the parameter space and make specific predictions concerning reprogramming dynamics which are broadly in line with experimental findings.
Kristian Thijssen (University of Oxford)
Title: Active nematics in confinement
The spontaneous emergence of collective flows is a generic property of active fluids. It is caused by the self-propelled, microscopic particles that drive the fluid out-of-equilibrium. In these active fluids, various collective flows can emerge including lamellar flow, vortex lattices and active turbulence with flow structures scaling by an intrinsic length scale many times larger than that of the individual particles. Here we consider active nematic in a channel where the channel height and the intrinsic active length scale compete, and show that their ratio governs a sequence of dynamical flow transitions. Understanding the mechanism of the flow transitions is of considerable importance in the design and control of active materials.
Priya Subramanian (University of Leeds)
Title: Formation and Spatial Localization of Phase Field Quasicrystals
The dynamics of many physical systems often evolve to asymptotic states that exhibit spatial and temporal variations in their properties such as density, temperature, etc. Regular patterns such as graph paper and honeycombs look the same when moved by a basic unit and/or rotated by certain special angles. They possess both translational and rotational symmetries giving rise to discrete spatial Fourier transforms. In contrast, an aperiodic crystal displays long range order but no periodicity. Such quasicrystals lack the lattice symmetries of regular crystals, yet have discrete Fourier spectra. We look to understand the minimal mechanism which promotes the formation of such quasicrystalline structures arising in diverse soft matter systems such as dendritic-, star-, and block co-polymers using a phase field crystal model. Direct numerical simulations combined with weakly nonlinear analysis highlight the parameter values where the quasicrystals are the global minimum energy state and help determine the phase diagram. By locating parameter values where multiple patterned states possess the same free energy (Maxwell points), we obtain states where a patch of one type of pattern (for example, a quasicrystal) is present in the background of another (for example, the homogeneous liquid state) in the form of spatially localized dodecagonal (in 2D) and icosahedral (in 3D) quasicrystals. In two dimensions, we compute several families of spatially localized quasicrystals with dodecagonal structure and investigate their properties as a function of the system parameters. The presence of such metastable localized quasicrystals is significant as they affect the dynamics of the soft matter crystallization process.
Livio Gibelli (University of Warwick)
Title: A mean field kinetic theory modelling of liquid-vapour flows a the micro/nano scale
In spite of the increasing number of experimental, theoretical and numerical studies, many aspects of non-equilibrium fluid flows involving phase change are still unclear with numerous unresolved issues.
The talk focuses on their kinetic theory modelling based on the Enskog-Vlasov equation.
This equation provides an approximate description of the microscopic behaviour of the fluid composed of spherical molecules interacting by Sutherland potential, but it has the capability of handling both the liquid and vapor phase, including the interface region.
As such, it is a useful bridge between the continuum approaches which fail to properly deal with the complexities of interfacial phenomena and the Molecular Dynamics (MD) simulations which require a huge computational effort.
After a brief overview of the basic elements of kinetic theory, the Enskog-Vlasov equation is presented and several applications are discussed.
These include the evaporation of a liquid film, the formation/breakage of liquid menisci in micro/nano-channels, and the nanodrop impact on a solid surface.
Bio: Livio Gibelli is research fellow in the School of Engineering at the University of Warwick. He received his PhD in applied mathematics from the Politecnico di Milano and, prior to his present position, he served as research fellow at the Politecnico di Torino, Politecnico di Milano, and the University of British Columbia.
His main research interests include
the kinetic theory modelling of non-equilibrium multiphase systems, the continuum description of slightly rarefied gas flows, the numerical methods for solving kinetic equations, and the mesoscopic modelling of social systems and crowd dynamics
Mark Hoefer (University of Colorado Boulder)
Title: Dispersive hydrodynamics: the mathematics and physics of nonlinear waves in dispersive media
Dispersive hydrodynamics--modeled by hyperbolic conservation laws with dispersive perturbation--has emerged as a unified mathematical framework for the description of multiscale nonlinear wave phenomena in dispersive media and accurately describes a plethora of physical systems. This talk will be a tour through some recent mathematical and experimental results in this growing field of research. Parallels and analogies to classical hydrodynamics will be presented such as the generation of shock waves subject to appropriate regularization and their description in terms of characteristics. In contrast, from the existence of expansion shocks to the generation of dissipative shock waves in a conservative medium, dispersive regularization also leads to a number of counterintuitive, perhaps bizarre, effects, which will also be described. To keep you engaged, this tour will include lots of video and animations of in-house experiments and simulations.
Yuji Kodama (The Ohio State University)
Title: KP Solitons in Shallow Water
The Kadomtsev–Petviashvili (KP) equation describes weakly dispersive and small amplitude waves propagating in a quasi-two-dimensional situation. Recently a large variety of exact soliton solutions of the KP equation has been found and classified. Those soliton solutions are localized along certain lines in a two-dimensional plane and decay exponentially everywhere else, and they are called line-soliton solutions. The classification is based on the far-field patterns of the solutions, which consist of a finite number of line-solitons. Each soliton solution is then defined by a point of the totally non-negative Grassmann variety, which can be parameterized by a unique derangement of the symmetric group of permutations.
In this talk, I will give a brief summary of our theory and discuss an application to the Mach reflection problem in shallow water, which has an important implication to tsunami amplification along the shore. The problem describes the resonant interaction of solitary waves appearing in the reflection of an obliquely incident wave onto a vertical wall. In particular, nonlinear interactions among small amplitude, obliquely propagating long waves on the surface of shallow water often generate web-like patterns. I discuss how line-soliton solutions of the KP equation can approximate such web-pattern in shallow water wave. The talk will be elementary and include many figures of the wave-patterns from real ocean.
Costanza Benassi (University of Northumbria Newcastle)
Title: Loop correlations in random wire models.
Random loop models appear in a great variety of situations in both the probability and mathematical physics literature. For a wide class of them, a striking conjecture has been proposed: in dimensions three and higher, there should be phases with macroscopic loops and the joint distribution of their lengths is expected to be a member of the Poisson-Dirichlet distribution family. We propose a loop model which is strictly related to O(N) classical spin systems and which is an extension of the random current representation of the Ising model. We prove that even loop correlations are given by Poisson-Dirichlet. Based on a joint work with D. Ueltschi, arXiv:1807.06564.
Michael Shearer (North Carolina State University)
Title: Granular flow: Particle Size Segregation and Constitutive Laws
The segregation of particles of different sizes can be achieved by vibration or by shear flow.
I describe a simple nonlinear conservation law that captures the main features of segregation under shear flow,
such as occurs in an avalanche. Predictions of the theory are tested against experiments in a shear flow. The partial differential equation has interesting mathematical properties that are explored through analysis, explicit solutions and numerical simulation. In the second half of the talk I introduce constitutive laws for two-dimensional time dependent flow that lead to linear well-posedness. The new theory, known as the CIDR rheology, specifies a yield condition and flow rule related to the mu(I) rheology, but including compressibility that is crucial to establishing well-posedness.
Thibault Congy (Northumbria University)
Title: Schrödinger equations and the universal description of dispersive shock wave structure
A Dispersive Shock Wave (DSW) is an expanding, modulated nonlinear wavetrain that connects two disparate hydrodynamic states (see Figure), and can be viewed as a dispersive counterpart to the dissipative, classical shock. DSWs have raised a lot of interest in the recent years, due to the growing recognition of their fundamental nature and ubiquity in physical applications: from shoaling tsunami waves and internal undular bores in the ocean and atmosphere, to optical shocks in laser beam propagation, quantum shocks in superfluids, and nonlinear spin wave propagation in magnetic thin films. Although powerful methods like the Whitham modulation theory are available to describe the modulation of these structures, the study of DSWs remains a challenging mathematical problems, especially in the context of non-integrable nonlinear wave equations.
Orestis Georgiou (Ultrahaptics)
Title: Touching the invisible: the emerging field of mid-air ultrasonic haptics
Gesture based control interfaces are becoming ubiquitous in our daily lives. We use gestures to interact with our appliances, smartphones, electronic devices and, more recently, virtual and augmented environments. The availability of these solutions has been made possible by a new generation of camera tracking devices. One drawback of gesture based interfaces is their lack of physicality, feedback, and sense of agency. Recent developments in ultrasound technologies have bridged this gap and made it possible to combine gesture interfaces with haptic feedback. In this presentation, I will discuss how this is achieved, describe some of the research taking place in this space, and also the commercial applications that are on the horizon.
Klaus Richter (University of Regensburg, Germany)
Title: Probing the Quantum Mechanics of Many-Body Chaos
The dynamics and spread of quantum information in complex many-body systems is presently attracting a lot of attention across various fields, ranging from cold atom physics via condensed quantum matter to high energy physics and quantum gravity. This includes questions of how a quantum system thermalises, phenomena like many-body localisation and non-classicality in many-body quantum physics. Here concepts such as the famous spin echo that are based on rewinding time provide a useful way to monitor complex quantum information dynamics. Central to these developments are so-called out-of time-order correlators (OTOCs). They presently receive particular attention as sensitive probes for chaos and the temporal growth of complexity in interacting systems.
We will address such phenomena using semiclassical path intergral techniques based on interfering Feynman paths, thereby bridging the classical and quantum many-body world. This enables us to compute echoes and OTOCs in terms of coherent sums and interprete the results in terms of multi-particle interference, thereby including entanglement and correlation effects. Moreover, on the numerical side we devise a semiclassical method for Bose-Hubbard systems far-out-of equilibrium that allows us to calculate many-body quantum interference on time scales far beyond the famous scrambling/Ehrenfest time.
Hilmar Gudmundsson (Northumbria)
Title: The relationship between bed and surface topography on glaciers and ice sheets
Glacier flow is an example of a gravity driven non-linear viscous flow at low Reynolds numbers. As a glacier flows over an undulating bed, the surface topography is modified in response. Some information about bed conditions is therefore contained in the shape of the surface and the surface velocity field. I will present theoretical and numerical work on how basal conditions on glaciers affect ice flow, and how one can obtain information about basal conditions through surface-to-bed inversion. I’ll give an overview over inverse methodology currently used in glaciology, and how satellite data is now routinely used to invert for bed properties of the Greenland and the Antarctic Ice Sheets.
Sirio Orozco Fuentes (University of Newcastle)
Title: : Patterning, segregation and differentiation in human embryonic stem cell colonies
The maintenance of the pluripotent state in human embryonic stem cells (hESCs) is highly crucial for their application in the laboratory as a tool for drug testing and the study of cell based therapies. Currently the selection of the best quality cells and colonies for propagation is done empirically in terms of their displayed features, such as a round nucleus, scant cytoplasm, prominent/abundant nucleoli and less intercellular spacing between the individuals in the bulk. Using image analysis and computational tools, we quantify these properties using phase contrast images of hESCs colonies of different sizes (0.1 - 1.1 mm2) during day 2, 3 and 4 of plating.
We identify their main characteristics such as the number of nearest neighbours, mean cell area, among other features. We discuss the mechanisms underlying the formation of these structures in vitro and explore, through a dynamical model in which the cells are represented as Voronoi tessellations of the space, how the cells might attain different levels of pluripotency and differentiate towards the three germs layers.
Riccardo Montalto (University of Milano)
Title: KAM theory for pure gravity water waves in finite depth
In this talk I will present some recent results concerning the existence and the stability of quasi-periodic solutions for the WAVES EQUATIONS ( 2D-Euler equation of an irrotational and incompressible fluid in an ocean with finite depth under the action of the gravity). After an overview of the classical methods used in the KAM theory for semilinear partial differential equations, I will focus on the method used to deal with fully nonlinear PDEs and in particular I will describe the KAM results obtained for the water waves equation. The main difficulties are:
1) the fully nonlinear nature of the gravity water waves equations (the highest order x-derivative appears in the nonlinear term but not in the linearization at the origin)
2) the linear frequencies grow only in a sublinear way at infinity.
In order to overcome the small divisors problem, the proof is obtained by a Nash Moser iteration. The key point is to solve the linearized PDE at any approximate solution. This requires to combine perturbation theory and Pseudo differential calculus.
Gandalf Lechner (Cardiff University)
Title: Quantum-mechanical backflow and scattering theory
"Backflow is the phenomenon that the probability current of a quantum particle on the line can flow in the direction opposite to its momentum. This talk will revisit this effect in the context of potential scattering in quantum mechanics. It turns out backflow is universal in the sense that it occurs in every potential (in a large class), and has always bounded spatial extent. On a mathematical level, these results are proven by establishing (lower) bounds on the spectra of certain integral operators. These general investigations are complemented with concrete examples and numerics."
Ian Strachan (University of Glasgow)
Title: Miura transformations from Novikov algebras
For the KdV equation, the Muira map transforms the second Hamiltonian structure into the first Hamiltonian structure. Multicomponent generalization of KdV’s bi-Hamiltonian structure have been known for decades – they date back to the work of Gelfand and Dorfman, and Balinskii and Novikov – and are defined in terms of algebraic structures known as Novikov algebras. The corresponding Muira map for these structures was constructed by Balinskii and Novikov only in the case where the algebra is commutative. In this talk a construction will be presented which solves the problem of the constructive of these maps in general.
Daniel Ueltschi (University of Warwick)
Title: From condensed matter physics to probability theory
The primary goal of condensed matter physics is to understand the behaviour of electrons in solids. The basic
laws are well understood, but the large number of interacting particles makes it challenging. A popular approach
is to introduce simple models and to use the setting of statistical mechanics. I will review quantum spin systems
and their stochastic representations in terms of random permutations and random loops. I will also describe the
*universal* behaviour that is common to loop models in dimensions 3 and more.
Vincenzo Vitagliano (Keio University)
Title: Topological defects, deformed lattices and spontaneous symmetry breaking
External conditions have a dramatic impact on the way symmetry breaking occurs. I will review some recent (and some less recent) results of symmetry breaking in curved spacetime. Flirting with the contemporary interest toward 2D engineered material, I will then move on potential applications on geometrically deformed lattices. In a curved background, the natural expectation is that curvature works towards the restoration of internal symmetries. I will show instead that, for topological defects, the competing action of the locally induced curvature and boundary conditions generated by the non-trivial topology allows configurations where symmetries can be spontaneously broken close to the core.
Anna Concas (University of Cagliari)
Title: A spectral method for ''bipartizing'' a network and detecting a large anti-community
Relations between discrete quantities such as people, genes, or streets can be described by networks, which consist of nodes that are connected by edges. Network analysis aims to identify important nodes in a network and to uncover the structural properties of a network. A network is said to be bipartite if its nodes can be subdivided into two nonempty sets such that there are no edges between nodes in the same set. It is a difficult task to determine the closest bipartite network to a given network.
In this talk, I will describe how a given network can be approximated by a bipartite one by solving a sequence of fairly simple optimization problems. The proposed algorithm also produces a node permutation which makes the possible bipartite nature of the initial adjacency matrix evident and identifies the two sets of nodes. It will be also showed how the same procedure can be used to detect the presence of a large anti-community in a network and to identify it.
Michael Overton (Courant Institute, New York)
Titel: Stability Optimisation for Polynomials and Matrices
Suppose that the coefficients of a monic polynomial or entries of a square matrix depend affinely on parameters, and consider the problem of minimising the root radius (maximum of the moduli of the roots) or root abscissa (maximum of their real parts) in the polynomial case and the spectral radius or spectral abscissa in the matrix case.
These functions are not convex and they are typically not locally Lipschitz near minimisers. We first address polynomials, for which some remarkable analytical results are available in one special case, and then consider the more general case of matrices, focusing on the static output feedback problem arising in control of linear dynamical systems. We also briefly discuss some spectral radius optimisation problems arising in the analysis of the transient behaviour of a Markov chain and the design of
smooth surfaces using subdivision algorithms. Finally, time permitting, we discuss optimisation of pseudospectra of matrices.
Cornelis van der Mee (University of Cagliari)
Title: Reflectionless Solutions for Square Matrix Nonlinear Schroedinger equation with Vanishing Boundary Conditions
After a quick review of the direct and inverse scattering theory of the focusing Zakharov-Shabat system with symmetric nonvanishing boundary conditions, we derive the reflectionless solutions of the 2 + 2 matrix NLS equation with vanishing boundary conditions and four different symmetries by using the Marchenko theory. Since the Marchenko integral kernel has separated variables, the matrix triplet method - consisting of representing the Marchenko integral kernel in a suitable form - allows us to find the exact expressions of the reflectionless solutions in terms of a triplet of matrices. Moreover, since these exact expressions contain matrix exponentials and matrix inverses, computer algebra can be used to "unpack" and graph them. Finally, it is remarkable that these solutions are also verified by direct substitution in the 2 + 2 NLS equation.
This is a joint work with Francesco Demontis (University of Cagliari, Italy) and Alyssa Ortiz (University of Colorado at Colorado Springs, USA).
Traveling edge waves in photonic graphene
Photonic graphene, namely an honeycomb array of optical waveguide, has attracted much interest in the optics community. In recent experiments it was shown that introducing edges and suitable waveguides in the direction of propagation, unidirectional edge wave propagation at optical frequencies occurs in photonic graphene. The system is described analytically by the lattice nonlinear Schrodinger (NLS) equation with a honeycomb potential and a pseudo-magnetic field. In certain parameter regimes, these edge waves were found to be nearly immune to backscattering, and thus exhibit the hallmarks of (Floquet) topological insulators.
This talk addresses the linear and nonlinear dynamics of traveling edge waves in photonic graphene, using a tight-binding model derived from the lattice NLS equation. Two different asymptotic regimes are discussed, in which the pseudo-magnetic field is respectively assumed to vary rapidly and slowly. In the presence of nonlinearity, nonlinear edge solitons can exist due to the balance between dispersion and nonlinearity; these edge solitons appear to be immune to backscattering when the dispersion relation is topologically nontrivial. A remarkable effect of topology in bounded photonic graphene will also be demonstrated: the edge modes are found to exhibit strong transmission (reflection) around sharp corners when the dispersion relation is topologically nontrivial (trivial).
Fabio Briscese (Northumria University Newcastle)
Theoretical foundations of the Schroedinger method for the formation of large scale structures in the Universe
It has been shown that the formation of large scale structures (LSS) in the universe can be described in terms of a Schroedinger-Poisson system. This procedure, known as Schroedinger method, has no theoretical basis, but it is intended as a mere tool to model the N-body dynamics of dark matter halos which form LSS. Furthermore, in this approach the ``Planck constant” in the Schroedinger equation is just a free parameter. In this seminar I will discuss a possible derivation of the Schroedinger method which is based on the stochastic quantization introduced by Nelson, and on the Calogero conjecture. Moreover, I will discuss how the Calogero conjecture allows to estimate the value of the effective Planck constant.
Oleg Kirillov (Northumbria University Newcastle)
Diabolical and exceptional points in the families of non-Hermitian matrices
We consider a complex perturbation of multiparameter families of real symmetric and Hermitian matrices and study unfolding of conical singularities of eigensurfaces at diabolical points into new singular surfaces such as the conical wedge of Wallis. As a physical application, singularities of dispersion surfaces in crystal optics are studied. Asymptotic formulas for the metamorphoses of these surfaces depending
on properties of a crystal are established and discussed. Singular axes for general crystals with weak absorption and chirality are found. An explicit condition distinguishing the absorbtion-dominated and chirality-dominated crystals is established in terms of components of the inverse dielectric tensor. Finally, we turn to the question of approximate computation of the geometric phase along a path surrounding either DP or EP-set in the parameter space by means of the perturbation of eigenvectors at the degeneracies
Andrea Pizzoferrato (University of Warwick)
Lower current large deviations for zero-range processes on a ring
Non-equilibrium statistical mechanics is a broad field of research covering many different phenomena. In this context Interacting Particle Systems (IPS) are a well established class of minimal models, where particles jump on a lattice according to local jump rates. This talk considers one particular IPS called the Zero Range Process (ZRP), which is characterized by jump rates depending on the number of particles on the departure site only. Remarkably, for certain choices of the transition rate function, the model exhibits “condensation”, that is a finite fraction of the particles of the system concentrates on the same site. We focus on totally asymmetric dynamics with periodic boundary conditions in one dimension, and study the large deviations of the particle current. Lower large deviations can be realized by phase separated states of high and low density regions, which may degenerate into condensed profiles for condensing ZRPs. We establish a dynamic phase transition related to this crossover and derive the rate function for the current for a large class of ZRPs. The results presented can be found in more detail here: https://arxiv.org/abs/1611.03729. This is a joint work with Paul Chleboun and Stefan Grosskinsky.
Francesco Demontis (University of Cagliari)
Soliton solutions for the classical continuous Heisenberg ferromagnet equation
Many nonlinear differential equations can be solved via the Inverse Scattering Transform (IST). In this talk, after a brief introduction of the IST, we present a rigorous formulation of the IST for the classical continuous Heisenberg ferromagnet (HF) equation. This formulation is based on a new triangular representation for the Jost solutions, which in turn allows to establish the asymptotic behaviour of the scattering data for large values of the spectral parameter. A new, general, explicit multi-soliton solution formula, amenable to computer algebra, is obtained by means of the matrix triplet method, producing all the soliton solutions for the HF equation and allowing their classification and description.
This talk is based on a joint work with M. Sommacal and S. Lombardo (Northumbria University, Newcastle) and C. van der Mee and F. Vargiu (University of Cagliari).
Mario Angelelli (University of Salento)
If it is complex, make it simplex: A tropical approach to statistical physics
The concepts of partition function and free energy lie at the root of statistical physics and are now pervasive in many branches of sciences. This talk is meant to explore some algebraic and geometric aspects of these fundamental objects, with emphasis on their role in the connection of micro- and macro-physics. Firstly, some geometric realizations of these connections will be briefly presented. Then, main attention will be paid to the issues of composition and dominance, which are pivotal in statistical physics and will be expressed through tropical algebras. A basic introduction on these structures will be given in order to discuss the tropical limit of statistical models. Concrete examples will be provided, with particular focus on highly (exponentially) degenerate models and their physical manifestations (residual entropy, limiting temperatures). This scheme is formalized in a micro-macro correspondence, where microphysics is related to a macroscopic description within the composition rules given by tropical structures. Some physical consequences of this correspondence are discussed, i.e. relations with ultrametrics, non-equilibrium and tropical equilibrium, dependence on the choice of ground energy.
Antonio Moro (Northumbria University Newcastle)
Dressing networks: towards an integrability approach to collective and complex phenomena
A large variety of real world systems can be naturally modelled by networks, i.e. graphs whose nodes represent the components of a system linked (interacting) according to specific statistical rules. A network is realised by a graph typically constituted of a large number of nodes/links. Fluid and magnetic models in Physics are just two among the many classical examples of systems which can be modelled by using simple or complex networks. In particular "extreme" conditions (thermodynamic regime), networks, just like fluids and magnets, exhibit a critical collective behaviour intended as a drastic change of state due to a continuous change of thermodynamic parameters.
Using an approach to thermodynamics, recently introduced to describe a general class of van der Waals type models and magnetic systems in mean field approximation, we analyse the integrable structure of corresponding networks and use the theory of nonlinear conservation laws to provide an analytical description of the system outside and inside the critical region.
Vladimir Novikov (Loughborough University)
Multi-dimensional integrable systems: from dispersionless to soliton equations
In this talk we will consider the problem of studying, detecting and classifying integrable 2+1-dimensional equations. Our approach is based on the observation that dispersionless limits of integrable systems in 2 + 1 dimensions possess infinitely many multi-phase solutions coming from the so-called hydrodynamic reductions. We develop a novel perturbative approach to the classification problem of dispersive equations. Based on the method of hydrodynamic reductions, we first classify integrable quasilinear systems which may (potentially) occur as dispersionless limits of soliton equations in 2 + 1 dimensions. To reconstruct dispersive deformations, we require that all hydrodynamic reductions of the dispersionless limit are inherited by the corresponding dispersive counterpart. This procedure leads to a complete list of integrable third and fifth order equations, which generalize the examples of Kadomtsev-Petviashvili, Veselov-Novikov and Harry Dym equations as well as integrable Davey-Stewartson type equations, some of which are apparently new. We also extend the technique to differential-difference and fully discrete integrable systems in 3D.
Arseni Goussev (Northumbria University Newcastle)
Rotating Gaussian wave packets in weak external potentials
Among many motivations to study the time evolution of quantum matter-wave packets two are particularly noteworthy. First, localized wave packets provide the most natural tool for investigating the correspondence between quantum and classical motion. Indeed, while the center of a propagating wave packet traces a trajectory, a concept essential in classical mechanics, its finite spatial extent makes quantum interference effects possible. Second, wave packets may be used as basis functions. That is, any initial state of a quantum system can be represented as a superposition of a number, finite or infinite, of localized wave packets. Thus, an analytical understanding of how each individual wave packet moves through space offers a way to quantitatively describe the time-evolution of an arbitrary, often complex, initial state. Despite a large body of literature on quantum wave packet dynamics, the subject is by no means exhausted.
In a recent paper [1], Dodonov has addressed the time evolution of nonrelativistic two-dimensional Gaussian wave packets with a finite value of mean angular momentum (MAM). The value is the sum of the "external" MAM, related to the center of mass motion, and the "internal" MAM, resulting from the rotation of the wave packet around its center of mass. Among many interesting features of such wave packets is the effect of initial shrinking of packets with strong enough coordinate-momentum correlation.
In my talk, motivated by Ref. [1], I will consider the propagation of a localized two- or three-dimensional rotating Gaussian wave packet in the presence of a weak external potential. The particular focus will be on the time evolution of the internal MAM of the moving wave packet. I will derive, using a semiclassical approximation of the eikonal type, an explicit formula that gives the value of the internal MAM as a function of the propagation time, parameters of the initial wave packet and the external potential. An example physical scenario, in which a two-dimensional particle traverses a tilted ridge barrier, will be considered in full detail. In particular, it will be shown how an initially uncorrelated, rotation-free wave packet may, upon a collision with the potential barrier, acquire a finite internal MAM, and how the magnitude and direction of the MAM are determined by the aspect ratio and orientation of the incident wave packet.
[1] V. V. Dodonov. "Rotating quantum Gaussian packets," J. Phys. A: Math. Theor. 48, 435303 (2015).
Gregory Morozov (University of the West of Scotland)
Exactly solvable Hill's equations
Remy Dubertrand (University of Liege)
Scattering theory for walking droplets in the presence of obstacles
Walking droplets that are sustained on the surface of a vibrating liquid, have attracted considerable attention during the past decade due to their remarkable analogy with quantum wave-particle duality. This was initiated by the pioneering experiment by Y. Couder and E. Fort in 2006, which reported the observation of a diffraction pattern in the angular resolved profile of droplets that propagated across a single slit obstacle geometry. While the occurrence of this wave-like phenomenon can be qualitatively traced back to the coupling of the droplet with its associated surface wave, a quantitative framework for the description of the surface-wave-propelled motion of the droplet in the presence of confining boundaries and obstacles still represents a major challenge. This problem is all the more stimulating as several experiments have already reported clear effects of the geometry on the dynamics of walking droplets.
Here we present a simple model inspired from quantum mechanics for the dynamics of a walking droplet in an arbitrary geometry. We propose to describe its trajectory using a Green function approach. The Green function is related to the Helmholtz equation with Neumann boundary conditions on the obstacle(s) and outgoing conditions at infinity.
For a single slit geometry our model is exactly solvable and reproduces some of the features observed experimentally. It stands for a promising candidate to account for the presence of boundaries in the walker’s dynamics.
Martin Brinkmann (Universitat de Saarlandes)
Wettability controls fluid transport in particulate and permeable media
While the impact of pore shape and pore size distributions on immiscible displacement has been widely studied, only a few previous experiments and simulations have addressed the effect of pore-scale wettability. In this talk, I will present examples of two and three dimensional experimental model system where both geometry and wettability are controllable factors. Optical microscopy and x-ray tomography are employed to monitor and track the distribution of fluids down to the
level of single pores. To gain further insight into the underlying pore-scale processes, we have developed a stochastic rotation dynamic model for multi-phase flows that account for wall wettability. This allows us to assess and validate experiments of fluid transport in Hele-Shaw cells with cylindrical obstacles or random packings of spherical beads.
Benoit Huard (Northumbria University Newcastle)
Title: Periodic solutions in delay equations with application to glucose regulation rhythms
Delay-differential equations represent a class of dynamical systems which offer an enhanced modelling of physical and biological processes through the incorporation of time delays. These typically represent mechanisms, physical or physiological, which incur a delayed response within the system. The resulting dynamical system is infinite-dimensional and one can very rarely obtain closed-form exact solutions. Nonetheless, characterising bifurcations in these models leads to precise conditions ensuring that the system enters an oscillatory regime.
In this talk, I will be focusing on the description of the so-called ultradian rhythms which are crucial for the appropriate regulation of blood glucose. Two models, with one and two delays respectively, are studied to characterise the periodic solutions and the effect of deficiencies (Type 1 and Type 2 diabetes) on the production of an oscillatory regime. Pathways for restoring these oscillations are discussed. Approximate periodic solutions are obtained using a perturbative analysis. Finally, a new model making use of a state-dependent delay is introduced to assess the efficacy of emergency mechanisms under high glucose levels.
Alexander Veselov (University of Loughborough)
Markov triples: arithmetic, geometry and dynamics
Markov triples are the integer solutions of the Markov equation x^2+y^2+z^2=3xyz, which surprisingly appeared in many areas of mathematics: initially in classical number theory, but more recently in hyperbolic and algebraic geometry, the theory of Teichmueller spaces, Frobenius manifolds and Painleve equations.
I am planning to explain some of these magical relations and properties of Markov triples, including recent results about their growth found jointly with K. Spalding.
Jonathan Halliwell (Imperial College London)
How the Quantum Universe became Classical
Quantum theory has been spectacularly successful in its description of the microscopic realm and there is not one shred of experimental evidence that suggests that its basic structure is incorrect in any way. Yet despite being the fundamental theory of matter, it is not the theory of the ordinary everyday world of our experience and indeed some of its features such as the existence of superposition states and entanglement defy intuition in a truly profound way. This is because the macroscopic world and our everyday experience is best described by the classical mechanics of Newton and the laws of thermodynamics, theories utterly different to quantum theory. This then leads to a very interesting question. If macroscopic objects are made of atoms, and atoms are described by quantum theory, how do large collections of atoms come to be described by classical physics? In brief, how does classical physics emerge from quantum theory at sufficiently large scales? In this talk I outline in simple terms how this transition from quantum to classical physics takes place, with an emphasis on simple physical ideas and pictures, and not elaborate mathematics. It includes a simple account of the decoherent histories approach to quantum theory which is particularly suited to this task.
Sven Gnutzmann (University of Nottingham)
Title: Quantum signatures of chirality in chaotic quantum systems
We revisit the model of two coupled spins by Peres and Feingold. Thirty years ago this was a paradigm system of quantum chaos. We show that this model can be used as a paradigm to understand universal features in the density of states for the novel symmetry classes in The Altland-Zirnbauer's ten-fold symmetry classification. Reducing the system to invariant subspaces reveals that the same classical system may belong to different symmetry classes depending on the spin quantum numbers. Moreover a nonzero topological quantum numbers can be found in various cases. By varying coupling constants the system makes a transition from integrable to chaotic and we show numerically that the repulsion of the quantum energy eigenvalues close to E=0 is consistent with Gaussian random matrix predictions for the corresponding combination of symmetry class and topological quantum number. No proper disorder average is necessary in the sense that we can keep the classical dynamics fixed and average over different quantum representations (spin quantum numbers).
Mon 12th - 3pm - Andrew Hone (University of Kent)
Title: Peakonomics
The Camassa-Holm equation was originally proposed as a model for shallow water waves. It is also an integrable partial differential equation which appeared in earlier work of Fokas and Fuchssteiner on hereditary symmetries, and as well as having a (bi-)Hamiltonian formulation, it can also be interpreted as a geodesic flow on the group of diffeomorphisms with respect to a suitable Sobolev metric. However, perhaps its most interesting feature is that, in the dispersionless case, its solitons are weak solutions - "peakons" - with a discontinuous derivative at isolated points.
After a brief review of peakons in integrable systems, a model for urban growth due to Krugman is presented, which was proposed as a mechanism for the formation of edge cities (that is to say, regions of concentrated economic activity). We show that Krugman's model admits exact measure-valued solutions, with an associated quantity, called the market potential, being given by a superposition of peakons with two different widths. Consideration of the dynamical system for this peakon interaction leads to a conjecture on the form of a global attractor in Krugman's model.
Richard Bertram (Florida State University)
Title: How Simple Concepts from Dynamics Can Drive Biological Experiments
Sometimes knowing a few concepts from dynamics can take you a long way. In this presentation I will demonstrate this claim by describing some research on the origin of pulsatile insulin secretion from pancreatic islets of Langerhans. This research has largely been driven by mathematical insights, which include some well known and very useful concepts about dynamical systems. I hope to relay the message that mathematical theory, computer simulations, and experimental studies can be combined very effectively to increase our understanding of complex biological phenomena.
Matteo Sommacal (Northumbria University)
Title: Linear stability analysis of integrable partial differential equations
Analytical methods of the theory of integrable partial differential equations (PDEs) in 1+1 dimensions have been successfully applied to investigate a number of wave propagation models of physical interest. This talk shows how to address the issue of linear stability of wave solutions by means of these methods. By imitating the standard steps followed when dealing with non-integrable equations, we show how the linear stability of solutions of integrable PDEs can be effectively analysed by using their Lax representation. The most relevant application of this scheme is the analysis of the background continuous wave solution and of the conditions for the existence of rogue waves. The talk is based on work done in collaboration with Antonio Degasperis, University of Rome La Sapienza, Rome, Italy and Sara Lombardo, Northumbria University, Newcastle upon Tyne, UK.
Francesco Giglio (Northumbria University)
Title: Integrable Nematics
Liquid crystals are considered to be, after gases, liquids and solids, the forth natural state of matter on Earth, as terrestrial free standing plasmas have not been yet observed.
Nematic liquid crystals (nematics in short) are characterised by the fact that they occur in two macroscopic phases: isotropic and anisotropic phases. While in the former each molecule is oriented randomly in space, in the latter molecules tend to align along special directions giving rise to a macroscopic anisotropy of the material.
Inspired by a recent formulation of Thermodynamics based on the theory of nonlinear conservation laws, we propose a novel approach to the Statistical Mechanics of nematics that relies on the study of differential identities for the partition function.
We show how exact equations of state of a discrete version of the Maier Saupe model can obtained from the solution of a suitable initial value problem for a partial differential equation.
Aim of the talk is to illustrate the approach and discuss the rich Thermodynamics features encoded in equations of state so obtained.
Rodrigo Ledesma-Aguilar (Northumbria University)
Title: Snap evaporation on smooth topographies
The evaporation of droplets on solid surfaces is important for a broad range of applications that include ink-jet printing, surface cooling, and nano- and micro-structure assembly. Despite its apparent simplicity, the precise configuration of an evaporating droplet on a solid surface has proven notoriously difficult to predict and control. This is because droplet evaporation typically proceeds as a ‘stick-slip’ sequence, which is a combination of pinning and de-pinning events of the droplet edge dominated by the static friction, or ‘pinning’, caused by microscopic structure of the solid surface. Here we show how smooth, pinning-free, solid surfaces of non-planar topography give rise to a different process which we dub snap evaporation. During snap evaporation the morphology of an evaporating droplet follows a reproducible sequence of steps, where the liquid-gas interface is quasi-statically reduced by mass diffusion until it undergoes an out-of-equilibrium snap. Such events are triggered by bifurcations of the equilibrium droplet configuration promoted by the underlying surface topography. Because the evolution of droplets during snap evaporation is controlled by the geometry of the solid, and not by microscopic surface roughness, our ideas are amenable to programmable surfaces which manage evaporation in heat and mass transfer applications.
Adam Bridgewater (Northumbria University Newcastle)
Title: Mathematical investigation of diabetically impaired ultradian oscillations
in the glucose-insulin regulatory system
The accuracy of the oscillatory nature of the glucose and insulin blood levels constitutes an important indicator of healthy regulation [4]. In this contribution, we study a mathematical model which incorporates two physiological delays, as well as parameters representing diabetic deficiencies, in order to investigate the impact of a fault in the glucose utilisation on the production of ultradian oscillations in the glucose-insulin system. The delays represent the hepatic glucose production and the time necessary for the release of insulin, and have been shown to be at least partly responsible for the oscillatory nature of the regime [2,3]. A numerical study of the non-linear mathematical model is performed to characterise the onset of the oscillations, and perturbations of the periodic solutions are used to investigate the amplitude and frequency of the oscillations. Through use of linear stability analysis and bifurcation theory, pathways to restoring appropriate cyclic regulation are mathematically described [1]. Our goal is to provide measurable indicators of deficiency in the system and a more thorough description of the contribution of insulin treatments to the reintroduction of an oscillatory behaviour which is crucial for the design of a control algorithm which could then be implemented into an insulin control system.
[1] Huard, B., Bridgewater, A., Angelova, M. (2017), J Theor Biol, 418, pp.66-76.
[2] Li, J., Kuang, Y., Mason, C. (2006), J Theor Biol 242(3), 722-735.
[3] Sturis, J. et al (1991), AM J Physiol-Endoc M 260(5), E801-E809.
[4] Tolic, I. et al (2000), J Theor Biol 207(3), 361-375.
Cosimo Gorini (University of Regensburg)
Title: Magnetoconductance in (curved) topological insulator nanowires
We investigate quantum transport in 3D topological insulator nanowires in external electric and magnetic fields. The wires host topologically non-trivial surface states wrapped around an insulating bulk, and a magnetic field along the wire axis leads to Aharonov-Bohm oscillations of the conductance. Such oscillations have been observed in numerous systems and signal surface transport, though alone cannot prove its topological nature. Furthermore, it is not known how they are affected by the wire specific geometry - never perfectly cylindrical as in standard theoretical models.
We thus focus on two issues: (i) An accurate modelling of surface transport in gated, strained HgTe nanowires, accompanying experimental measurements performed by our collaborators; (ii) A theoretical study of magneto-conductance through shaped (tapered, curved) nanowires. The nanowire non-constant radius leads to novel quantum transport phenomena. Notably, it implies a competition between effects due to quantum confinement
and to a spatially varying enclosed magnetic flux, as well as offering the possibility of studying quantum Hall physics in curved space.
Sergiy Shelyag (Northumbria University)
Title: Numerical Models for solar plasmas
Due to their gravitational stratification and rotation, solar and stellar plasmas represent a complicated and interesting object for numerical modelling. Orders-of-magnitude variations in thermodynamic and magnetic parameters over short spatial scales impose additional stiffness into the system of partial differential equations describing the plasma behaviour. A variety of physical effects, which affect plasma parameters and act on a large range of spatial and temporal scales, require robustness and computational effectiveness of the numerical methods employed in solar plasma modelling. In my presentation I will cover my own experience in development of numerical models for simulations of solar plasmas. I will describe numerical methods for simulations of waves, flows and radiation in solar plasmas and present an overview of the results, applicable in solar physics.
Ciro Semprebon (Northumbria University)
Title: Numerical simulations of ternary multiphase-multicomponent systems
Systems containing two or more fluid phases and one gas phase are of special practical interest. Oil lubricated surfaces offer a variety of unique properties such as superior non-wetting, ice-repellent and robust drag-reduction. In engines, collision merged droplets of oil and water droplets can increase the effective burning rate. Despite the growing number of applications and the advancing experimental techniques, accurate and flexible models that can predict complex interface dynamics of ternary systems with significant density ratio are lacking. In this talk I will introduce two novel free energy Lattice Boltzmann models and show applications in Liquid Infused Substrates and collisions of immiscible drops.
Kuo-Long Pan (University of Leeds)
Title: On the impact of binary droplets with identical and distinct liquids
Collision between droplets plays a critical role in various subjects of science and technology. For instance, the collisions of oil droplets are the essential elements in liquid-fueled burning devices such as aircraft/rocket combustors and automobile engines. They are the key processes underlying spray combustion, concerning the interactions between different groups of droplets or between droplets and the confinement, which dominate the subsequent distributions of mass, momentum, and energy that consequently determine the behaviors and performance of the burners. The typical outcomes of droplet collisions can be categorized into coalescence, bouncing, temporary coalescence and separation followed by satellite droplets, and splattering, with increase of impact inertia relative to surface tension. In addition to the significance in aerospace applications, the impact dynamics of droplets are important in many other fields as well, e.g. raindrop formation and aerosol phenomena, ink-jet printing technology, firefighting strategy, nuclear power generation, spray painting and coating techniques, etc. To understand the mechanisms and further control the collision outcomes, we have studied the fundamental structures in terms of experimental, analytical, and computational approaches, associated with various physical schemes and length scales. Comprehension of the principles can be extended to other intriguing areas specifically micro/nano multiphase fluidics that has been fervidly investigated recently due to its immediate relevance to biological/medical techniques and industry. In addition to droplets made of identical liquid, we have also studied the collisions between two droplets made of different liquids. In this talk I will introduce some of these studies and discuss the elementary mechanisms.
Biography
Prof. Kuo-Long Pan graduated from National Tsing Hua University in 1994 (B.S.) and 1996 (MS) in Power Mechanical Engineering, and received PhD from Princeton University in 2004 in Mechanical and Aerospace Engineering. Since 2013 he has been a full professor in Department of Mechanical Engineering, National Taiwan University. His research interests lie in fluid physics, combustion and energy, computational fluid dynamics and propulsion. He is an Associate Fellow of American Institute of Aeronautics and Astronautics (AIAA) and a member of American Physical Society (APS). He has received several awards including a Best Paper Award of AIAA (2005)
Fabian Maucher (University of Durham)
Title: Two Examples of Self-organization
Self-organization represents one of the most striking phenomena of complex systems. In this talk I present two
examples of self-organization. The rst example presents dynamics of spiral waves governed by reaction-diusion
equations. There are a range of chemical, physical and biological excitable media that support spiral wave vortices.
Examples include the Belousov-Zhabotinsky redox reaction, the chemotaxis of slime mould and action potentials
in cardiac tissue. There are types of reaction-diusion equations that give rise to strong short-ranged repulsive
interactions between vortex strings [1]. A Biot-Savart construction to initialize a given knot as a vortex string in
the FitzHugh-Nagumo equations is presented [see Fig. 1(a)]. The subsequent evolution gives rise to self-organized
untangling of vortex strings [2]. Light-propagation in media with competing nonlocal nonlinearities represents the
second example of self-organization [3]. Such system can be realized in a gas of thermal alkali atoms. Apart from
spatial soliton formation, the dierent length scales of the nonlocality can give rise to lamentation and subsequent
self-organised hexagonal lattice formation in the beam prole [see Fig. 1(b)], akin to the superuid-supersolid phase
transition in Bose-Einstein condensates.
Oleg Chalykh (University of Leeds)
Title: Dunkl-Cherednik operators and quantum Lax pairs
I will explain a direct link between Dunkl operators and quantum Lax pairs for the elliptic Calogero--Moser systems. This produces Lax pairs with a spectral parameter, which in the classical limit reduce to the well-known Krichever's Lax pair and its analogues for other root systems. Time permitting, I will outline a similar construction for the elliptic Ruijsenaars-Schneider system and its variants. The Dunkl operators in that case are replaced by their q-analogues, also known as Cherednik operators.
Alessia Annibale (King's College London)
Title: Cell reprogramming modelled as transitions in a hierarchy of cell cycles
We construct a model of cell reprogramming (the conversion of fully differentiated cells to a state of pluripotency, known as induced pluripotent stem cells, or iPSCs) which builds on key elements of cell biology i.e. cell cycles and cell lineages. Although reprogramming has been demonstrated experimentally, much of the underlying processes governing cell fate decisions remain unknown. This work aims to bridge this gap by modelling cell types as a set of hierarchically related dynamical attractors representing cell cycles. Stages of the cell cycle are characterised by the configuration of gene expression levels, and reprogramming corresponds to triggering transitions between such configurations. Two mechanisms were found for reprogramming in a two level hierarchy: cycle specific perturbations and a noise-induced switching. The former corresponds to a directed perturbation that induces a transition into a cycle state of a different cell type in the potency hierarchy (mainly a stem cell) whilst the latter is a priori undirected and could be induced, e.g., by a (stochastic) change in the cellular environment. These reprogramming protocols were found to be effective in large regimes of the parameter space and make specific predictions concerning reprogramming dynamics which are broadly in line with experimental findings.
Kristian Thijssen (University of Oxford)
Title: Active nematics in confinement
The spontaneous emergence of collective flows is a generic property of active fluids. It is caused by the self-propelled, microscopic particles that drive the fluid out-of-equilibrium. In these active fluids, various collective flows can emerge including lamellar flow, vortex lattices and active turbulence with flow structures scaling by an intrinsic length scale many times larger than that of the individual particles. Here we consider active nematic in a channel where the channel height and the intrinsic active length scale compete, and show that their ratio governs a sequence of dynamical flow transitions. Understanding the mechanism of the flow transitions is of considerable importance in the design and control of active materials.
Priya Subramanian (University of Leeds)
Title: Formation and Spatial Localization of Phase Field Quasicrystals
The dynamics of many physical systems often evolve to asymptotic states that exhibit spatial and temporal variations in their properties such as density, temperature, etc. Regular patterns such as graph paper and honeycombs look the same when moved by a basic unit and/or rotated by certain special angles. They possess both translational and rotational symmetries giving rise to discrete spatial Fourier transforms. In contrast, an aperiodic crystal displays long range order but no periodicity. Such quasicrystals lack the lattice symmetries of regular crystals, yet have discrete Fourier spectra. We look to understand the minimal mechanism which promotes the formation of such quasicrystalline structures arising in diverse soft matter systems such as dendritic-, star-, and block co-polymers using a phase field crystal model. Direct numerical simulations combined with weakly nonlinear analysis highlight the parameter values where the quasicrystals are the global minimum energy state and help determine the phase diagram. By locating parameter values where multiple patterned states possess the same free energy (Maxwell points), we obtain states where a patch of one type of pattern (for example, a quasicrystal) is present in the background of another (for example, the homogeneous liquid state) in the form of spatially localized dodecagonal (in 2D) and icosahedral (in 3D) quasicrystals. In two dimensions, we compute several families of spatially localized quasicrystals with dodecagonal structure and investigate their properties as a function of the system parameters. The presence of such metastable localized quasicrystals is significant as they affect the dynamics of the soft matter crystallization process.
Livio Gibelli (University of Warwick)
Title: A mean field kinetic theory modelling of liquid-vapour flows a the micro/nano scale
In spite of the increasing number of experimental, theoretical and numerical studies, many aspects of non-equilibrium fluid flows involving phase change are still unclear with numerous unresolved issues.
The talk focuses on their kinetic theory modelling based on the Enskog-Vlasov equation.
This equation provides an approximate description of the microscopic behaviour of the fluid composed of spherical molecules interacting by Sutherland potential, but it has the capability of handling both the liquid and vapor phase, including the interface region.
As such, it is a useful bridge between the continuum approaches which fail to properly deal with the complexities of interfacial phenomena and the Molecular Dynamics (MD) simulations which require a huge computational effort.
After a brief overview of the basic elements of kinetic theory, the Enskog-Vlasov equation is presented and several applications are discussed.
These include the evaporation of a liquid film, the formation/breakage of liquid menisci in micro/nano-channels, and the nanodrop impact on a solid surface.
Bio: Livio Gibelli is research fellow in the School of Engineering at the University of Warwick. He received his PhD in applied mathematics from the Politecnico di Milano and, prior to his present position, he served as research fellow at the Politecnico di Torino, Politecnico di Milano, and the University of British Columbia.
His main research interests include
the kinetic theory modelling of non-equilibrium multiphase systems, the continuum description of slightly rarefied gas flows, the numerical methods for solving kinetic equations, and the mesoscopic modelling of social systems and crowd dynamics
Mark Hoefer (University of Colorado Boulder)
Title: Dispersive hydrodynamics: the mathematics and physics of nonlinear waves in dispersive media
Dispersive hydrodynamics--modeled by hyperbolic conservation laws with dispersive perturbation--has emerged as a unified mathematical framework for the description of multiscale nonlinear wave phenomena in dispersive media and accurately describes a plethora of physical systems. This talk will be a tour through some recent mathematical and experimental results in this growing field of research. Parallels and analogies to classical hydrodynamics will be presented such as the generation of shock waves subject to appropriate regularization and their description in terms of characteristics. In contrast, from the existence of expansion shocks to the generation of dissipative shock waves in a conservative medium, dispersive regularization also leads to a number of counterintuitive, perhaps bizarre, effects, which will also be described. To keep you engaged, this tour will include lots of video and animations of in-house experiments and simulations.
Yuji Kodama (The Ohio State University)
Title: KP Solitons in Shallow Water
The Kadomtsev–Petviashvili (KP) equation describes weakly dispersive and small amplitude waves propagating in a quasi-two-dimensional situation. Recently a large variety of exact soliton solutions of the KP equation has been found and classified. Those soliton solutions are localized along certain lines in a two-dimensional plane and decay exponentially everywhere else, and they are called line-soliton solutions. The classification is based on the far-field patterns of the solutions, which consist of a finite number of line-solitons. Each soliton solution is then defined by a point of the totally non-negative Grassmann variety, which can be parameterized by a unique derangement of the symmetric group of permutations.
In this talk, I will give a brief summary of our theory and discuss an application to the Mach reflection problem in shallow water, which has an important implication to tsunami amplification along the shore. The problem describes the resonant interaction of solitary waves appearing in the reflection of an obliquely incident wave onto a vertical wall. In particular, nonlinear interactions among small amplitude, obliquely propagating long waves on the surface of shallow water often generate web-like patterns. I discuss how line-soliton solutions of the KP equation can approximate such web-pattern in shallow water wave. The talk will be elementary and include many figures of the wave-patterns from real ocean.
Costanza Benassi (University of Northumbria Newcastle)
Title: Loop correlations in random wire models.
Random loop models appear in a great variety of situations in both the probability and mathematical physics literature. For a wide class of them, a striking conjecture has been proposed: in dimensions three and higher, there should be phases with macroscopic loops and the joint distribution of their lengths is expected to be a member of the Poisson-Dirichlet distribution family. We propose a loop model which is strictly related to O(N) classical spin systems and which is an extension of the random current representation of the Ising model. We prove that even loop correlations are given by Poisson-Dirichlet. Based on a joint work with D. Ueltschi, arXiv:1807.06564.
Michael Shearer (North Carolina State University)
Title: Granular flow: Particle Size Segregation and Constitutive Laws
The segregation of particles of different sizes can be achieved by vibration or by shear flow.
I describe a simple nonlinear conservation law that captures the main features of segregation under shear flow,
such as occurs in an avalanche. Predictions of the theory are tested against experiments in a shear flow. The partial differential equation has interesting mathematical properties that are explored through analysis, explicit solutions and numerical simulation. In the second half of the talk I introduce constitutive laws for two-dimensional time dependent flow that lead to linear well-posedness. The new theory, known as the CIDR rheology, specifies a yield condition and flow rule related to the mu(I) rheology, but including compressibility that is crucial to establishing well-posedness.
Thibault Congy (Northumbria University)
Title: Schrödinger equations and the universal description of dispersive shock wave structure
A Dispersive Shock Wave (DSW) is an expanding, modulated nonlinear wavetrain that connects two disparate hydrodynamic states (see Figure), and can be viewed as a dispersive counterpart to the dissipative, classical shock. DSWs have raised a lot of interest in the recent years, due to the growing recognition of their fundamental nature and ubiquity in physical applications: from shoaling tsunami waves and internal undular bores in the ocean and atmosphere, to optical shocks in laser beam propagation, quantum shocks in superfluids, and nonlinear spin wave propagation in magnetic thin films. Although powerful methods like the Whitham modulation theory are available to describe the modulation of these structures, the study of DSWs remains a challenging mathematical problems, especially in the context of non-integrable nonlinear wave equations.
Orestis Georgiou (Ultrahaptics)
Title: Touching the invisible: the emerging field of mid-air ultrasonic haptics
Gesture based control interfaces are becoming ubiquitous in our daily lives. We use gestures to interact with our appliances, smartphones, electronic devices and, more recently, virtual and augmented environments. The availability of these solutions has been made possible by a new generation of camera tracking devices. One drawback of gesture based interfaces is their lack of physicality, feedback, and sense of agency. Recent developments in ultrasound technologies have bridged this gap and made it possible to combine gesture interfaces with haptic feedback. In this presentation, I will discuss how this is achieved, describe some of the research taking place in this space, and also the commercial applications that are on the horizon.
Klaus Richter (University of Regensburg, Germany)
Title: Probing the Quantum Mechanics of Many-Body Chaos
The dynamics and spread of quantum information in complex many-body systems is presently attracting a lot of attention across various fields, ranging from cold atom physics via condensed quantum matter to high energy physics and quantum gravity. This includes questions of how a quantum system thermalises, phenomena like many-body localisation and non-classicality in many-body quantum physics. Here concepts such as the famous spin echo that are based on rewinding time provide a useful way to monitor complex quantum information dynamics. Central to these developments are so-called out-of time-order correlators (OTOCs). They presently receive particular attention as sensitive probes for chaos and the temporal growth of complexity in interacting systems.
We will address such phenomena using semiclassical path intergral techniques based on interfering Feynman paths, thereby bridging the classical and quantum many-body world. This enables us to compute echoes and OTOCs in terms of coherent sums and interprete the results in terms of multi-particle interference, thereby including entanglement and correlation effects. Moreover, on the numerical side we devise a semiclassical method for Bose-Hubbard systems far-out-of equilibrium that allows us to calculate many-body quantum interference on time scales far beyond the famous scrambling/Ehrenfest time.
Hilmar Gudmundsson (Northumbria)
Title: The relationship between bed and surface topography on glaciers and ice sheets
Glacier flow is an example of a gravity driven non-linear viscous flow at low Reynolds numbers. As a glacier flows over an undulating bed, the surface topography is modified in response. Some information about bed conditions is therefore contained in the shape of the surface and the surface velocity field. I will present theoretical and numerical work on how basal conditions on glaciers affect ice flow, and how one can obtain information about basal conditions through surface-to-bed inversion. I’ll give an overview over inverse methodology currently used in glaciology, and how satellite data is now routinely used to invert for bed properties of the Greenland and the Antarctic Ice Sheets.
Sirio Orozco Fuentes (University of Newcastle)
Title: : Patterning, segregation and differentiation in human embryonic stem cell colonies
The maintenance of the pluripotent state in human embryonic stem cells (hESCs) is highly crucial for their application in the laboratory as a tool for drug testing and the study of cell based therapies. Currently the selection of the best quality cells and colonies for propagation is done empirically in terms of their displayed features, such as a round nucleus, scant cytoplasm, prominent/abundant nucleoli and less intercellular spacing between the individuals in the bulk. Using image analysis and computational tools, we quantify these properties using phase contrast images of hESCs colonies of different sizes (0.1 - 1.1 mm2) during day 2, 3 and 4 of plating.
We identify their main characteristics such as the number of nearest neighbours, mean cell area, among other features. We discuss the mechanisms underlying the formation of these structures in vitro and explore, through a dynamical model in which the cells are represented as Voronoi tessellations of the space, how the cells might attain different levels of pluripotency and differentiate towards the three germs layers.
Riccardo Montalto (University of Milano)
Title: KAM theory for pure gravity water waves in finite depth
In this talk I will present some recent results concerning the existence and the stability of quasi-periodic solutions for the WAVES EQUATIONS ( 2D-Euler equation of an irrotational and incompressible fluid in an ocean with finite depth under the action of the gravity). After an overview of the classical methods used in the KAM theory for semilinear partial differential equations, I will focus on the method used to deal with fully nonlinear PDEs and in particular I will describe the KAM results obtained for the water waves equation. The main difficulties are:
1) the fully nonlinear nature of the gravity water waves equations (the highest order x-derivative appears in the nonlinear term but not in the linearization at the origin)
2) the linear frequencies grow only in a sublinear way at infinity.
In order to overcome the small divisors problem, the proof is obtained by a Nash Moser iteration. The key point is to solve the linearized PDE at any approximate solution. This requires to combine perturbation theory and Pseudo differential calculus.
Gandalf Lechner (Cardiff University)
Title: Quantum-mechanical backflow and scattering theory
"Backflow is the phenomenon that the probability current of a quantum particle on the line can flow in the direction opposite to its momentum. This talk will revisit this effect in the context of potential scattering in quantum mechanics. It turns out backflow is universal in the sense that it occurs in every potential (in a large class), and has always bounded spatial extent. On a mathematical level, these results are proven by establishing (lower) bounds on the spectra of certain integral operators. These general investigations are complemented with concrete examples and numerics."
Ian Strachan (University of Glasgow)
Title: Miura transformations from Novikov algebras
For the KdV equation, the Muira map transforms the second Hamiltonian structure into the first Hamiltonian structure. Multicomponent generalization of KdV’s bi-Hamiltonian structure have been known for decades – they date back to the work of Gelfand and Dorfman, and Balinskii and Novikov – and are defined in terms of algebraic structures known as Novikov algebras. The corresponding Muira map for these structures was constructed by Balinskii and Novikov only in the case where the algebra is commutative. In this talk a construction will be presented which solves the problem of the constructive of these maps in general.
Daniel Ueltschi (University of Warwick)
Title: From condensed matter physics to probability theory
The primary goal of condensed matter physics is to understand the behaviour of electrons in solids. The basic
laws are well understood, but the large number of interacting particles makes it challenging. A popular approach
is to introduce simple models and to use the setting of statistical mechanics. I will review quantum spin systems
and their stochastic representations in terms of random permutations and random loops. I will also describe the
*universal* behaviour that is common to loop models in dimensions 3 and more.
Vincenzo Vitagliano (Keio University)
Title: Topological defects, deformed lattices and spontaneous symmetry breaking
External conditions have a dramatic impact on the way symmetry breaking occurs. I will review some recent (and some less recent) results of symmetry breaking in curved spacetime. Flirting with the contemporary interest toward 2D engineered material, I will then move on potential applications on geometrically deformed lattices. In a curved background, the natural expectation is that curvature works towards the restoration of internal symmetries. I will show instead that, for topological defects, the competing action of the locally induced curvature and boundary conditions generated by the non-trivial topology allows configurations where symmetries can be spontaneously broken close to the core.
Anna Concas (University of Cagliari)
Title: A spectral method for ''bipartizing'' a network and detecting a large anti-community
Relations between discrete quantities such as people, genes, or streets can be described by networks, which consist of nodes that are connected by edges. Network analysis aims to identify important nodes in a network and to uncover the structural properties of a network. A network is said to be bipartite if its nodes can be subdivided into two nonempty sets such that there are no edges between nodes in the same set. It is a difficult task to determine the closest bipartite network to a given network.
In this talk, I will describe how a given network can be approximated by a bipartite one by solving a sequence of fairly simple optimization problems. The proposed algorithm also produces a node permutation which makes the possible bipartite nature of the initial adjacency matrix evident and identifies the two sets of nodes. It will be also showed how the same procedure can be used to detect the presence of a large anti-community in a network and to identify it.
Michael Overton (Courant Institute, New York)
Titel: Stability Optimisation for Polynomials and Matrices
Suppose that the coefficients of a monic polynomial or entries of a square matrix depend affinely on parameters, and consider the problem of minimising the root radius (maximum of the moduli of the roots) or root abscissa (maximum of their real parts) in the polynomial case and the spectral radius or spectral abscissa in the matrix case.
These functions are not convex and they are typically not locally Lipschitz near minimisers. We first address polynomials, for which some remarkable analytical results are available in one special case, and then consider the more general case of matrices, focusing on the static output feedback problem arising in control of linear dynamical systems. We also briefly discuss some spectral radius optimisation problems arising in the analysis of the transient behaviour of a Markov chain and the design of
smooth surfaces using subdivision algorithms. Finally, time permitting, we discuss optimisation of pseudospectra of matrices.
Cornelis van der Mee (University of Cagliari)
Title: Reflectionless Solutions for Square Matrix Nonlinear Schroedinger equation with Vanishing Boundary Conditions
After a quick review of the direct and inverse scattering theory of the focusing Zakharov-Shabat system with symmetric nonvanishing boundary conditions, we derive the reflectionless solutions of the 2 + 2 matrix NLS equation with vanishing boundary conditions and four different symmetries by using the Marchenko theory. Since the Marchenko integral kernel has separated variables, the matrix triplet method - consisting of representing the Marchenko integral kernel in a suitable form - allows us to find the exact expressions of the reflectionless solutions in terms of a triplet of matrices. Moreover, since these exact expressions contain matrix exponentials and matrix inverses, computer algebra can be used to "unpack" and graph them. Finally, it is remarkable that these solutions are also verified by direct substitution in the 2 + 2 NLS equation.
This is a joint work with Francesco Demontis (University of Cagliari, Italy) and Alyssa Ortiz (University of Colorado at Colorado Springs, USA).